A rigid body is studied in a series of finitely separated positions, in order to determine those points which lie on a special locus (a sphere, circle, plane, line, or cylinder). Equations governing these special points are derived and their numerical evaluation is discussed. Several numerical examples are presented. In a companion paper [21], these results are applied to the synthesis of spatial linkages, and special motions (e.g., planer and spherical) are incorporated into the general theory presented herein.